We can simulate data from a normal distribution as follows:

set.seed(144)
x1 <- rnorm(n= 1000, 0, 1)
x2 <- rnorm(n=1000, 3, 4)

data <- as.data.frame(cbind(x1, x2))

You could also sample form a multivariate normal distributions, but we won’t do that here.

require(MASS)
Sigma <- matrix(c(7,5,5,9),2,2)
data_mv <- as.data.frame(mvrnorm(n = 1000, rep(0, 2), Sigma))

require(DT, warn.conflicts = F)
datatable(data, options = list(pageLength = 5))

require(reshape, warn.conflicts = F)
data <- melt(data)
datatable(data, options = list(pageLength = 5))

require(plotly)

p <- ggplot(data, aes(x = value, fill = variable)) + 
  geom_density(alpha=.25) + 
  ggtitle("Normal densities")

ggplotly(p)

So two beautiful normal distributions.

This doesn’t relate to the normal density stuff, but we can take the inverse of functions:

\[ \begin{align} g_i(\mu_{ij}) &= \nu_{i} + \lambda_i \eta_j \\ \mu_{ij} &= g_i^{-1}(\nu_{i} + \lambda_i \eta_j). \end{align} \]

These relate to the Generalized Linear Factor Analysis Framework, originally proposed by